The CLT for Markov chains with a countable state space embedded in the space lp

We find necessary and sufficient conditions for the CLT for Markov chains with a countable state space embedded in the space lp for p⩾1. This result is an extension of the uniform CLT over the family of indicator functions in Levental (Stochastic Processes Appl. 34 (1990) 245–253), where the result is equivalent to our case p=1. A similar extension for the uniform CLT over a family of possibly unbounded functions in Tsai (Taiwan. J. Math. 1(4) (1997) 481–498) is also obtained.